Page 198 - Modern Control of DC-Based Power Systems

162 Modern Control of DC-Based Power Systems

5.5 BACKSTEPPING BASED CONTROL

In the following subsections the reader is introduced to the theory
behind Backstepping [51]. In a nutshell it can be described as a systematic
way to construct a control law which is based on Lyapunov function
while partitioning the system and extending the controlled system size
with each step. The reader will encounter a short description of what
Backstepping is, which will be followed by an overview of the stability
concept according to Lyapunov, which is the basis for the Control
Lyapunov Function (CLF). The Backstepping concept was successfully
applied to the control of DC DC converters in [52] supplying a resistive
load. In [53] and [54] an adaptive design with time-varying loads is pre-
sented, although with the restriction that load change is slow compared
to the converter dynamics.
To the authors’ best knowledge, the concept of Backstepping was
applied for the first time on the stabilization of CPLs and multiple parallel
LRCs in [55] and [56].
In Section 5.5.1 the theory behind Backstepping is explained; this
includes the stability concept of Lyapunov in 5.5.1.1 and CLFs in 5.5.1.2.
In Section 5.5.2 the procedure of Backstepping is explained. The reader
who is familiar with those theoretical concepts will find in Section 5.5.3
the application of the Backstepping theory in combination with the aug-
mented Kalman filter of Section 5.4.1.3 for decoupling the network on
the ISPS circuit.

5.5.1 Theory Behind Backstepping
This section gives an overview of Lyapunov theory in general and the
integrator Backstepping technique. Furthermore, this basic principle will
be just referred to as Backstepping throughout this book.
The peculiarity with nonlinear systems is that stability is no longer a
global property; it is restricted to certain trajectories of the system.
Therefore, many different stability criteria exist for specific classes of nonlin-
ear systems but the theory is not as developed as for linear systems yet. A very
often used stability concept is based on the theory introduced by Lyapunov.
According to Lyapunov, the stability of the system can be ensured by
finding a so-called Lyapunov function which fulfills the criteria defined in
the direct method of Lyapunov. However, in practice there are cases where
it proves to be very difficult to find such a solution for complex systems.